Let \(p \in M\) . In this proof we combine the observations made earlier in this section. The proofs of the four sub-claims are given below.
(REF )
By Lemma REF  we know that \(T(p)\)  is a smooth hyper-surface of \(M\)  whose tangent space is normal to the vector field \(\nu (q)=Y_1(q)-Y_2(q)\)  for \(q \in T(p)\) . Since \(Y_1(q)\ne Y_2(q)\)  and \(\Vert Y_1(q)\Vert _g=\Vert Y_2(q)\Vert \)  we get from Cauchy-Schwarz inequality that
\(\langle Y_1(q), \nu (q)\rangle >0\quad \text{ and } \quad \langle Y_2(q), \nu (q)\rangle \rangle <0.\) 
Thus \(Y_1(q)\)  and \(Y_2(q)\)  hit \(T(p)\)  from different sides. If \(q \in T(p)\cap \partial M\)  and these surfaces are tangential to each other at \(q\)  we arrive in a contradiction: Since \(\nu (q)\)  is normal to both \(T(T(p))\)  and \(T\partial M\)  we can without loss of generality assume that \(Y_2(q)\)  is inward pointing at \(q\) . This is not possible since the geodesic related to \(Y_2(q)\) , that connects \(p\)  to \(q\) , is contained in \(M\) . Thus by equation (REF ) \(Y_2(q)\)  is also outward pointing which is not possible.
(REF )
By Proposition REF  we know that the Hausdorff dimension of the conjugate cut locus \(Q(p)\)  does not exceed \(n-2\) . If we can prove the same for the set \(L(p)\)  of a-typical cut points the claim (REF ) follows from Lemma REF .
Recall that \(L(p)\subset ({\text{cut}}(p)\setminus Q(p))\)  is the set of points in \(M\)  that can be connected to \(p\)  with more than two distance minimizing geodesics of \(M\) . Let \(q \in L(p)\)  and define \(k_p(q)\in \mathbf {N}\)  to be the number of distance minimizing geodesics from \(p\)  to \(q\) . Then we choose vectors \(w_1,\ldots ,w_{k_p(q)} \in M_p\)  and their respective neighborhoods \(U_i \in T_pN\)  such that for each \(i \in \lbrace 1,\ldots ,k_p(q)\rbrace \)
\(\exp _p(w_i)=q, \quad \hbox{ and } \exp _{p}\colon U_i \rightarrow \tilde{V}\) 
is a diffeomorphism on some open set \(\tilde{V} \subset N\) . Let \(Y_i\)  for \(i \in \lbrace 1,\ldots ,k_p(q)\rbrace \)  be the distance vector fields related to the \(U_i\)  and \(p\) . Then we define a collection of smooth functions
\(\rho _{ij}\colon \tilde{V} \rightarrow {\bf R}, \quad \rho _{ij}(x)=\Vert Y_i(x)\Vert _{g}-\Vert Y_j(x)\Vert _{g}, \quad i,j \in \lbrace 1,\ldots ,k_p(q)\rbrace .\) 
By the proof of Lemma REF  it holds that the sets
\(K_{ij}:=\rho _{ij}^{-1}\lbrace 0\rbrace , \: \hbox{ for } i<j\) 
are smooth hyper-surfaces of \(N\)  that contain \(q\) . Also by [1]} it holds that the sets
\(K_{i,j,k}:=K_{ik}\cap K_{jk}, \quad \hbox{ for } i<j<k\) 
are smooth submanifolds of \(N\)  of co-dimension two. Next we set \(K(q):=\bigcup _{i<j<k}K_{i,j,k}\)  and claim that
\(L(p)\cap \tilde{V} = K(q)\cap M.\) 
Since the sets \(K_{i,j,k}\)  are smooth sub-manifolds of dimension \(n-2\)  their Hausdorff dimension is also \(n-2\) . Thus the equation (REF ) and Lemma REF  imply that Hausdorff dimension of \(L(p)\)  does not exceed \(n-2\) .
Finally we verify the equation (REF ). If \(x \in L(p)\cap \tilde{V}\)  it holds there are at least three distance minimizing geodesics of \(M\)  connecting \(p\)  to \(x\) . Thus there are \(1\le i<j<k\le k_{p}(q)\)  so that
\(\Vert Y_i(x)\Vert _g=\Vert Y_j(x)\Vert _g=\Vert Y_k(x)\Vert _g=d_M(p,x),\) 
which yields
\(\rho _{ik}(x)=\rho _{jk}(x)=0,\quad \hbox{and $x \in K_{i,j,k} \subset K(q) $}.\) 
If \(x \in K(q)\cap M\)  then \(x \in K_{i,j,k}\cap M\)  for some \(i<j<k\) . Thus by the proof of Lemma REF  it holds that there are at least three distance minimizing geodesics of \(M\)  connecting \(p\)  to \(x\) . Therefore \(x \in L(p)\cap \tilde{V}\) .
(REF )
Since we can write the cut locus of \(p\)  as a disjoint union
\({\text{cut}}(p)=T(p)\cup L(p) \cup Q(p)\) 
the parts (REF ) and (REF ) in conjunction with Lemma REF  yield the claim of part (REF ).
(REF )
Since \(\partial M\)  is a smooth hyper-surface of a \(n\) -dimensional Riemannian manifold \(M\)  we have by part (REF ) that \(T(p)\cap \partial M\)  is a smooth sub-manifold of dimension \(n-2\) , thus it has the Hausdorff-dimension \(n-2\) . Also by part (REF ) we know that the Hausdorff dimension of \(L(p)\cap Q(p)\)  does not exceed \(n-2\) . We have proven that the Hausdorff dimension of the closed set
\({\text{cut}}(p)\cap \partial M\) 
does not exceed \(n-2\) . Since the boundary of \(M\)  has the Hausdorff-dimension \(n-1\)  it follows that \(\partial M \setminus {\text{cut}}(p)\)  is open and dense in \(\partial M\) . The density claim follows from the observation that by Lemma REF  the set \({\text{cut}}(p)\cap \partial M\)  cannot contain any open subsets of \(\partial M\)  as their Hausdorff dimension is \(n-1\) .

